2. Opposite Numbers (additive inverse): is a number that when added to a given number yields 0. The opposite number for any number x is -x. Note that x may be positive or negative.

(-5) + 5 = 0

x + (-x) = 0

On the number line, the numbers 5 and -5 are located at the same distance from zero. We say that the numbers 5 and -5 are opposites. We can also say that -5 is opposite to 5, and 5 is opposite to -5 on the number line.

3. Addition of integers:

Addition of a positive number is taken as a movement to the RIGHT of the number line

Addition of a negative number is taken as a movement to the LEFT of the number line

(-3) + 4

2 + 3

(-1) + (-3)

2 + (-5)

0 + (-4)

4. Rule for addition:

If the signs of the integers being added are the same, the sum has the same sign as the integers and we add the absolute values of the integers

For any a > 0, and b > 0,

a + b = a + b; 3 + 5 = 8

(-a) + (-b) = -(a+b); (-3) + (-5) = – (3 + 5) = -8

If the signs of the integers being added are different,, the sum takes the sign of the integer with the greater absolute value and we find the difference of the absolute values of the integers.

1. Negative Numbers: we often across quantities that have opposite directions or meanings. For example, traveling due east and west, the rise and fall in price, profit and loss, and being above or below sea level. When we assign a certain meaning of quantity to be positive, a value of the quantity that has the opposite meaning may be considered as negative and is represented with a “-” sign.

Positive numbers: 1, 2, 3, 0.8, ⅜, …

Negative numbers: -1, -2, -3, -⅔, -0.0025, …

Meanings of positive and negative numbers: teaching on whiteboard in the class.

2. The (horizontal) number line:

How to draw a number line:

Draw a line, and mark the zero point on it;

Choose a unit length, e.g., 1 cm, to mark the points 1, 2, 3, … at equal unit intervals on the right of 0, and the points -1, -2, -3, … on the left of 0;

Draw an arrow at each end.

Inequality signs:

> greater than

< less than

>= greater or equal

<= less than or equal

On a horizontal number line:

All the positive numbers are to the right of 0

All the negative numbers are to the left of 0

Numbers are arranged in ascending (increasing) order from left to right

Every number is smaller than any number on its right and greater than any number on its left

3. Absolute Value: the absolute value of a number is the distance that number is from 0 on the number line. Both 3 and -3 are the same distance from 0. The absolute value of a number is never negative.

3*3 = 9, 9 is called the square of 3, we also say that 3 is the positive square root of 9.

2. Perfect Squares: the number 1, 4, 9, 16, 25, … whose square roots are whole numbers are called perfect squares. We can find the square root of a perfect square by using prime factorization. Teaching on whiteboard in the class.

3. Cube Roots

2*2*2 = 8, 8 is called the cube of 2, we also say that 2 is the cube root of 8.

4. Perfect Cubes: the number 1, 8, 27, 64, … whose cube roots are whole numbers are called perfect cubes. We can find the cube root of a perfect cube by using prime factorization. Teaching on whiteboard in the class.

1. Least Common Multiple (LCM): the least common multiple of a group of numbers is the smallest positive integer which is divisible by all the numbers in the group.

The multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …

The multiple of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …

24 and 48 are the first two common multiples of 6 and 8. Since 24 is the least of all common multiples, we say the least common multiple (LCM) of 6 and 8 is 24.

2. Methods to find LCM

Using prime factorization. LCM is obtained by multiplying the highest power of each prime factor of the given numbers. Teaching on whiteboard in the class.

Continuous division. Teaching on whiteboard in the class.

Venn diagram. Teaching on whiteboard in the class.

3. Why LCM?

adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. Ex. 1/15 + 1/18 + 1/21